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2-natural transformation?
The Yoneda lemma for bicategories is a version of the Yoneda lemma that applies to bicategories, the most common algebraic sort of weak 2-category. It says that for any bicategory $C$, any object $x\in C$, and any pseudofunctor $F\colon C^{op}\to Cat$, there is an equivalence of categories
which is pseudonatural in $x$ and $F$, and which is given by evaluation at $1_x$, i.e. $\alpha\colon C(-,x)\to F$ maps to $\alpha_x(1_x)$.
For bicategories $A$ and $B$, $[A,B]$ denotes the bicategory of pseudofunctors, pseudonatural transformations, and modifications from $A$ to $B$. Note that it is a strict 2-category as soon as $B$ is.
In particular, the Yoneda lemma for bicategories implies that there is a Yoneda embedding for bicategories $C\to [C^{op},Cat]$ which is 2-fully-faithful?, i.e. an equivalence on hom-categories. Therefore, $C$ is equivalent to a sub-bicategory of $[C^{op},Cat]$. Since Cat is a strict 2-category, it follows that $C$ is equivalent to a strict 2-category, which is one form of the coherence theorem for bicategories. (Conversely, another form of the coherence theorem can be used to prove the Yoneda lemma; see below.)
A detailed proof of the bicategorical Yoneda lemma seems to be hard to find in the literature. (Anyone have references to add?)
An explicit proof, involving many diagrams, has been written up by Igor Baković and can be found here.
We will take it for granted that $[C^{op},Cat]$ is a well-defined bicategory; this is a basic fact having nothing to do with the Yoneda lemma. We also take it as given that “evaluation at $1_x$” functor
is well-defined and pseudonatural in $F$ and $x$; our goal is to prove that it is an equivalence. (Granted, these basic facts require a fair amount of verification as well.)
We will use part of the coherence theorem for pseudoalegbras?, which says that for a suitably well-behaved strict 2-monad $T$, the inclusion $T$-$Alg_{strict} \hookrightarrow Ps$-$T$-$Alg$ of the 2-category of strict $T$-algebras and strict $T$-morphisms into the 2-category of pseudo $T$-algebras and pseudo $T$-morphisms has a left adjoint, usually written as $(-)'$. Moreover, for any pseudo $T$-algebra $A$, the unit $A\to A'$ is an equivalence in $Ps$-$T$-$Alg$.
First, there is a 2-monad $T$ such that strict $T$-algebras are strict 2-categories, strict $T$-morphisms are strict 2-functors, pseudo $T$-algebras are unbiased bicategories?, and pseudo $T$-morphisms are pseudofunctors. By Mac Lane’s coherence theorem for bicategories, any ordinary bicategory can equally well be considered as an unbiased one. Thus, since $Cat$ is a strict 2-category, for any bicategory $C$ there is a strict 2-category $C'$ such that pseudofunctors $C\to Cat$ are in bijection with strict 2-functors $C'\to Cat$.
Now note that a pseudonatural transformation between two pseudofunctors (resp. strict 2-functors) $C\to D$ is the same as a single pseudofunctor (resp. strict 2-functor) $C\to Cyl(D)$, where $Cyl(D)$ is the bicategory whose objects are the 1-cells of $D$, whose 1-cells are squares in $D$ commuting up to isomorphism, and whose 2-cells are “cylinders” in $D$. Likewise, a modification between two such transformations is the same as a single functor (of whichever sort) $C\to 2Cyl(D)$, where the objects of $2Cyl(D)$ are the 2-cells of $D$, and so on. Therefore, $C'$ classifies not only pseudofunctors out of $C$, but transformations and modifications between them; thus we have an isomorphism
where $[A,B]_{strict,pseudo}$ denotes the 2-category of strict 2-functors, pseudonatural transformations, and modifications between two strict 2-categories. Thus we can equally well analyze the functor
given by evaluation at $1_x$. Here $\overline{C(-,x)}$ and $\overline{F}$ denote the strict 2-functors $(C')^{op}\to Cat$ corresponding to the pseudofunctors $C(-,x)$ and $F$ under the $(-)'$ adjunction. However, we also have a strict 2-functor $C'(-,x)$, and the equivalence $C\simeq C'$ induces an equivalence $C'(-,x)\simeq \overline{C(-,x)}$. Therefore, it suffices to analyze the functor
Now for any $A$ and $B$, we have an inclusion functor $[A,B]_{strict,strict} \to [A,B]_{strict,pseudo}$ where $[A,B]_{strict,strict}$ denotes the 2-category of strict 2-functors, strict 2-natural transformations, and modifications. This functor is bijective on objects and locally fully faithful. Moreover, the composite
is an isomorphism, by the Yoneda lemma for enriched categories?, in the special case of $Cat$-enrichment. Since
is fully faithful, if we can show that it is essentially surjective, then the 2-out-of-3 property for equivalences of categories will imply that the desired functor is an equivalence.
Here we at last descend to something concrete. Given $\alpha\colon C'(-,x)\to \overline{F}$, we have an obvious choice for a strict transformation for it to be equivalent to, namely $\beta$ whose components $\beta_y\colon C'(y,x)\to \overline{F}(y)$ is given by $f \mapsto \overline{F}(f)(a)$ where $a = \alpha_x(1_x)\in \overline{F}(x)$. Since $\alpha$ is pseudonatural, for any $f\colon y\to x$ in $C'$ we have an isomorphism
We then simply verify that these isomorphisms are the components of an (invertible) modification $\alpha\cong \beta$. This completes the proof.
An account of Morita equivalence as a corollary of the Yoneda lemma for bicategories